Given \(v_i\) , the variance \(v_{i+1}\)  is distributed as a suitable scaled noncentral chi-squared distribution [1]}. Therefore, we substitute \({\int _{t_{i}}^{t_{i+1}}\sqrt{v(t)}{\rm d}\widetilde{W}^\lambda _v(t)}\)  in (REF ) using (), ending up with:
\(\begin{aligned}x_{i+1}&=x_{i}+\int _{t_{i}}^{t_{i+1}}\left(r-\frac{1}{2}v(t)\right){\rm d}t+\frac{\rho }{\gamma }\left(v_{i+1}-v_{i}-\kappa \int _{t_{i}}^{t_{i+1}}\left(\bar{v}-v(t)\right){\rm d}t\right)\nonumber \\&+\sqrt{1-\rho ^2}\int _{t_{i}}^{t_{i+1}}\sqrt{v(t)}{\rm d}\widetilde{W}_x(t).\end{aligned}\) 
